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Mirrors > Home > MPE Home > Th. List > cnncvsmulassdemo | Structured version Visualization version GIF version |
Description: Derive the associative law for complex number multiplication mulass 11218 interpreted as scalar multiplication to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
cnncvsmulassdemo | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . . 4 ⊢ (ringLMod‘ℂfld) = (ringLMod‘ℂfld) | |
2 | 1 | cncvs 25059 | . . 3 ⊢ (ringLMod‘ℂfld) ∈ ℂVec |
3 | id 22 | . . . 4 ⊢ ((ringLMod‘ℂfld) ∈ ℂVec → (ringLMod‘ℂfld) ∈ ℂVec) | |
4 | 3 | cvsclm 25040 | . . 3 ⊢ ((ringLMod‘ℂfld) ∈ ℂVec → (ringLMod‘ℂfld) ∈ ℂMod) |
5 | 2, 4 | ax-mp 5 | . 2 ⊢ (ringLMod‘ℂfld) ∈ ℂMod |
6 | 1 | cnrbas 25056 | . . . 4 ⊢ (Base‘(ringLMod‘ℂfld)) = ℂ |
7 | 6 | eqcomi 2736 | . . 3 ⊢ ℂ = (Base‘(ringLMod‘ℂfld)) |
8 | cnfldex 21269 | . . . 4 ⊢ ℂfld ∈ V | |
9 | rlmsca 21080 | . . . 4 ⊢ (ℂfld ∈ V → ℂfld = (Scalar‘(ringLMod‘ℂfld))) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ℂfld = (Scalar‘(ringLMod‘ℂfld)) |
11 | cnfldmul 21274 | . . . 4 ⊢ · = (.r‘ℂfld) | |
12 | rlmvsca 21082 | . . . 4 ⊢ (.r‘ℂfld) = ( ·𝑠 ‘(ringLMod‘ℂfld)) | |
13 | 11, 12 | eqtri 2755 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘ℂfld)) |
14 | cnfldbas 21270 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
15 | 14 | eqcomi 2736 | . . . 4 ⊢ (Base‘ℂfld) = ℂ |
16 | 15 | eqcomi 2736 | . . 3 ⊢ ℂ = (Base‘ℂfld) |
17 | 7, 10, 13, 16 | clmvsass 25003 | . 2 ⊢ (((ringLMod‘ℂfld) ∈ ℂMod ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
18 | 5, 17 | mpan 689 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ‘cfv 6542 (class class class)co 7414 ℂcc 11128 · cmul 11135 Basecbs 17171 .rcmulr 17225 Scalarcsca 17227 ·𝑠 cvsca 17228 ringLModcrglmod 21046 ℂfldccnfld 21266 ℂModcclm 24976 ℂVecccvs 25037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-addf 11209 ax-mulf 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-subg 19069 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-subrng 20472 df-subrg 20497 df-drng 20615 df-lmod 20734 df-lvec 20977 df-sra 21047 df-rgmod 21048 df-cnfld 21267 df-clm 24977 df-cvs 25038 |
This theorem is referenced by: (None) |
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